Symplectic classification for universal unfoldings of $A_n$ singularities in integrable systems

TL;DR

This paper classifies integrable Hamiltonian systems with $A_n$ singularities using symplectic normal forms, revealing that all such systems are symplectically equivalent near these singularities and providing explicit invariants.

Contribution

It provides a complete semi-local and semi-global symplectic classification of $A_n$ singularities in integrable systems, including explicit invariants and normal forms.

Findings

All systems are symplectically equivalent near $A_n$ singularities.

Explicit semi-local and semi-global invariants are given by tuples of real-analytic functions.

Classification covers cases from $n=1$ to higher dimensions, including parabolic points and cuspidal tori.

Abstract

In the present paper, we obtain real-analytic symplectic normal forms for integrable Hamiltonian systems with $n$ degrees of freedom near singular points having the type ``universal unfolding of $A_n$ singularity'', $n\ge1$ (local singularities), as well as near compact orbits containing such singular points (semi-local singularities). We also obtain a classification, up to real-analytic symplectic equivalence, of real-analytic Lagrangian foliations in saturated neighborhoods of such singular orbits (semi-global classification). These singularities (local, semi-local and semi-global ones) are structurally stable. It turns out that all integrable systems are symplectically equivalent near their singular points of this type (thus, there are no local symplectic invariants). A complete semi-local (respectively, semi-global) symplectic invariant of the singularity is given by a tuple of $n-1$ (respectively $n-1+\ell$) real-analytic function germs in $n$ variables, where $\ell$ is the number of connected components of the complement of the singular orbit in the fiber. The case $n=1$ corresponds to non-degenerate singularities (of elliptic and hyperbolic types) of one-degree of freedom Hamiltonians; their symplectic classifications were known. The case $n=2$ corresponds to parabolic points, parabolic orbits and cuspidal tori, and the case $n\ge3$ -- to their higher-dimensional analogs.